A211219 Maximum value of sigma(x) * sigma(y) * sigma(z), where x + y + z = n.

1, 3, 9, 27, 36, 63, 84, 147, 196, 343, 336, 588, 576, 1008, 864, 1728, 1152, 2160, 1872, 2700, 2340, 4032, 2925, 5040, 4368, 6300, 5460, 9408, 6552, 11760, 10192, 14112, 10080, 21952, 12096, 18816, 18816, 24304, 16128, 30576, 20832, 32928, 26208, 33852

Offset: 3

Author

Paolo P. Lava, Apr 05 2012

Keywords

• nonn

Examples

			For n=79 the maximum product is reached when 24, 27 and 28 (24+27+28=79) are considered: sigma(24)*sigma(27)*sigma(28) = 60*40*56 = 134400.
For n=83 the maximum product is reached when 24, 24 and 35 (24+24+35=83) are considered: sigma(24)*sigma(24)*sigma(35) = 60*60*48 = 172800.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 3..500

Crossrefs

Cf. A211216-A211218.

Programs

Maple

with(numtheory);
A211219:=proc(i)
local a,b,c,d,m,n,s;
for n from 3 to i do
  s:=0; a:=0; b:=0; c:=n;
  while a<=floor(n/3) do
    while b<=floor((n-a)/2) do
      for m from 1 to 3 do d:=sigma(a)*sigma(b)*sigma(c); od;
      if d>s then s:=d; fi; b:=b+1; c:=c-1;
    od;
    a:=a+1; b:=a; c:=n-a-b;
  od;
  print(s);
od; end:
A211219(1000);

Mathematica

a[n_] := Max[Times @@ DivisorSigma[1, #]& /@ IntegerPartitions[n, {3}]]; Table[a[n], {n, 3, 100}] (* Jean-François Alcover, Dec 26 2013 *)